1998 Paper 3 Q14

Year: 1998
Paper: 3
Question Number: 14

Course: UFM Statistics
Section: Central limit theorem

Difficulty: 1700.0 Banger: 1500.0

estimation uniform distribution order statistics unbiased estimator variance comparison maximum likelihood method of moments expected value efficiency

Problem

A hostile naval power possesses a large, unknown number $N$ of submarines. Interception of radio signals yields a small number $n$ of their identification numbers $X_i$ ($i=1,2,...,n$), which are taken to be independent and uniformly distributed over the continuous range from $0$ to $N$. Show that $Z_1$ and $Z_2$, defined by $$ Z_1 = {n+1\over n} {\max}\{X_1,X_2,...,X_n\} \hspace{0.3in} {\rm and} \hspace{0.3in} Z_2 = {2\over n} \sum_{i=1}^n X_i \;, $$ both have means equal to $N$. Calculate the variance of $Z_1$ and of $Z_2$. Which estimator do you prefer, and why?

No solution available for this problem.

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Difficulty Rating: 1700.0

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Banger Rating: 1500.0

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Problem source

A hostile naval power possesses a large, unknown number $N$ of 
submarines. Interception of radio signals yields a small number $n$ 
of their identification numbers $X_i$ ($i=1,2,...,n$), which are taken
to be independent and uniformly distributed over the continuous range
from $0$ to $N$. Show that $Z_1$ and $Z_2$, defined by
$$
Z_1 = {n+1\over n} {\max}\{X_1,X_2,...,X_n\} 
\hspace{0.3in} {\rm and} \hspace{0.3in}
Z_2 = {2\over n} \sum_{i=1}^n X_i \;,
$$
both have means equal to $N$.
Calculate the variance of $Z_1$ and of $Z_2$. Which estimator
do you prefer, and why?

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